Lift and Unmask: Sparse Models by Lifting the Atomic Norm

Pablo Rademacher, Carlos A. Sing-Long

Session: General

Keywords: Signal reconstruction; signal recovery; Compressed Sensing; sparse signal recovery; atomic norm; atomic norm minimization

TL;DR: We introduce a convex optimization method to select sparse models consisting on masked atoms in atomic norm recovery

Abstract: Sparse signal recovery has become one of the preferred methods to recover signals from a set of incomplete linear measurements. This is due both to its appealing computational properties, as it involves solving a convex optimization problem, and its rigorous justification by the theory of Compressed Sensing. When the underlying signal is not sparse, but it is instead a sparse combination of elementary building blocks called atoms, the signal can be recovered by minimizing the atomic norm, i.e., the gauge associated to the convex hull of the atomic set. Although this approach has been successfully used in several applications, there is an implicit geometric constraint in this approach: only the atoms that are exposed points of the convex hull will be selected to represent the solution to atomic norm minimization. This can be an issue when the representation of the underlying signal is sparse when using all the atoms, but dense when using exposed ones. In this work, we propose an approach based on lifting that allows us to promote representations using atoms that are not exposed. Our method is based on convex optimization, preserving many of the computational benefits of atomic norm minimization. We present phase diagrams derived from a suitable signal model showing the benefits of using our approach.

Submission Number: 100